Unsquaring Numbers

Unsquaring Numbers
Objective To introduce the concept of square roots and the
use of the square-root key on a calculator.
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ePresentations
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Use exponential notation to name square
numbers, and explore the relationship
between square numbers and square roots. [Number and Numeration Goal 4]
Key Activities
Students investigate “unsquaring” numbers
without using the square-root key on a
calculator and use the square-root key to
test their answers. They explore properties
of square numbers and their square roots.
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
Playing Multiplication Top-It
(Extended-Facts Version)
Student Reference Book, p. 334
Math Masters, p. 493
per partnership: 4 each of number
cards 1–10 (from the Everything Math
Deck, if available), calculator
Students use their knowledge of
extended facts to form and compare
numbers.
Ongoing Assessment:
Recognizing Student Achievement
Key Vocabulary
Use Math Masters, page 493. unsquaring a number square root square-root key
Math Boxes 1 8
Materials
Math Journal 1, p. 23
Study Link 17
calculator overhead calculator (optional) 16 counters (optional)
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
ENRICHMENT
Comparing Numbers with Their Squares
Math Masters, p. 23
calculator
Students investigate the relationship between
numbers and their squares.
EXTRA PRACTICE
5-Minute Math
5-Minute Math™, p. 108
slate or paper
Students practice using the square root sign.
[Operations and Computation Goal 2]
Math Journal 1, p. 24
Students practice and maintain skills
through Math Box problems.
Study Link 1 8
Math Masters, p. 22
Students practice and maintain skills
through Study Link activities.
Advance Preparation
Familiarize yourself with the use of the square root key on your students’ calculators.
Teacher’s Reference Manual, Grades 4–6 pp. 74, 75, 79–83, 94–98, 267–269
52
Unit 1
Number Theory
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Getting Started
Mental Math and Reflexes
Math Message
Pose the following problems. Have students
write an expression as you describe the calculation.
Students are not expected to calculate the solution.
Answers may vary.
Find the numbers that make these statements true.
∗
=4
2
= 81
The sum of 9 and 8 9 + 8
7 less than 7 7 - 7
The quotient of 24 divided by 6 24 / 6
7 less than the product of 2 and 9 (2 ∗ 9) - 7
Double 8 and then add 0 more (2 ∗ 8) + 0
0 times the sum of 8 and 2 0 ∗ (8 + 2)
Study Link 1 7 Follow-Up
Have partners compare answers and resolve
differences.
8 less than the sum of 10 and 5 (10 + 5) - 8
3 more than triple 3 3 + (3 ∗ 3)
10 less than triple 10 (3 ∗ 10) - 10
Interactive whiteboard-ready
ePresentations are available at
www.everydaymathonline.com to
help you teach the lesson.
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
Algebraic Thinking Ask a volunteer to read the first problem.
Some number times some number equals 4. What numbers could
the placeholders in this problem represent? The factors of 4; 1 and
4; 2 Write the first problem on the board or a transparency,
replacing both placeholders with the letter n. Ask students what
number the letter n (the variable) represents. 2 Explain that the
variable can only represent one number if the number sentence is
true. Tell students that unknown numbers will be represented
using variables as the placeholders. Rewrite the second problem
using the variable m. m2 = 81 Ask a volunteer to read the problem.
m squared equals 81; m ∗ m equals 81 What is the number m
that makes this number sentence true? 9
▶ “Unsquaring” Numbers
WHOLE-CLASS
ACTIVITY
PROBLEM
PRO
PR
P
RO
R
OB
BLE
BL
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
VIN
ING
Begin this activity by explaining that solving problems like the
Math Message problems requires unsquaring a number. We
needed to undo the operation that squared the number. If students
square a number, they multiply it by itself to get the product.
Given the product of a squared number, they have to undo the
multiplication in order to identify the number that was squared.
4∗4=p
Square the number 4 to find p.
n ∗ n = 16
Unsquare the number 16 to find n.
Adjusting
the Activity
ELL
Have students use counters to build the
square array for 16. Note that when 16 is
unsquared, the result is the same as the
number of rows, or the number of columns,
of the original square.
AUDITORY
KINESTHETIC
TACTILE
VISUAL
The difference between squaring and unsquaring a number
Lesson 1 8
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Ask: What number, multiplied by itself, is equal to 289? Give
students a few minutes to find the number. They may use their
calculators if they wish.
After a few minutes, survey the class for their solution strategies.
Most students will have used one of the following approaches:
The random method: Some students might have tried various
numbers without using a system to guide their choices.
The “squeeze” method: Some students might have tried various
numbers, each time using the result to help select their next
choice. To unsquare 289, you might:
●
Try 10: 102 = 100; much less than 289
●
Try 20: 202 = 400; more than 289
Then try numbers between 10 and 20, probably closer to 20 than
to 10.
Suggested square numbers for students to “unsquare”
196 14
10,000 100
1,024 32
676 26
7,225 85
3,136 56
441 21
900 30
1,849 43
5,041 71
䉬
Try 17: 172 = 289; the answer is 17.
Endings and products: When students established an interval,
such as the interval from 10 to 20, some might have reasoned
that since 17 ends in 7, and 7 ∗ 7 = 49; then 17 should be the
next choice because 289 also ends in 9.
Tell students that when they unsquare a number, they have found
the square root of the number. What number squared is 64? 8 So
what is 64 unsquared? 8, because 8 ∗ 8 = 64 What is the square
root of 64? 8
Unsquaring Numbers
18
●
Give students a few more square numbers to unsquare. (See
margin.) Challenge them to use as few guesses as possible.
Time
LESSON
Try 18: 182 = 324; still too large, but closer.
If students mention using the square-root key on their calculator,
acknowledge that this is an efficient way of unsquaring a number,
but the focus on this portion of the lesson is to help them
understand the process of squaring and unsquaring numbers
before they use the calculator function.
Student Page
Date
●
You know that 62 ⫽ 6 ⴱ 6 ⫽ 36. The number 36 is called the square of 6. If you
unsquare 36, the result is 6. The number 6 is called the square root of 36.
1.
Unsquare each number. The result is its square root. Do not use the
on your calculator.
12
Example:
15
27
40
19
a.
b.
c.
d.
2.
2
2
2
2
key
▶ Finding the Square Root
12 .
15
⫽ 225 The square root of 225 is
.
⫽ 729 The square root of 729 is 27 .
⫽ 1,600 The square root of 1,600 is 40 .
⫽ 361 The square root of 361 is 19 .
2
⫽ 144
The square root of 144 is
of Numbers
(Math Journal 1, p. 23)
Allow partners a few minutes to complete Problems 1 and 2.
Survey the class for suggestions for checking the answers in
Problem 1. Most students will respond with the following
possibilities:
Which of the following are square numbers? Circle them.
576
794
1,044
4,356
6,400
5,770
List all factors of each square number. Make a factor rainbow to check your work. Then
fill in the missing numbers.
3.
49:
1 7 49
4.
64:
1
7
2
⫽ 49
7
The square root of 49 is
PARTNER
ACTIVITY
ELL
.
Multiply the square root of a number by itself.
1
5.
81:
6.
100:
1
2
3 9 27 81
2
8
4 8 16 32 64
4
9
2
⫽ 81
2
The square root of 81 is
5 10 20 25 50 100
The square root of 100 is
10
8
⫽ 64 The square root of 64 is
10
.
2
⫽ 100
9
.
.
Use the square-root key, for example
find the square root of a number.
, on the calculator to
To support English language learners, write the following on the
board: 82 = 8 ∗ 8 = 64. The square of 8 is 64. The square root of 64
is 8.
Math Journal 1, p. 23
54
Unit 1
Number Theory
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Game Master
Explain that in the same way that the class has used a calculator
to test the result of other computations, they will use a calculator
to test that they have accurately found the square-root of a
number. If available, use an overhead calculator to demonstrate
how to use the square-root function key. Model for students how to
test the answers in Problem 2. Emphasize the following points:
Name
Date
Time
1 2
4 3
Top-It Record Sheet
Round
>, <, =
Player 1
Player 2
Sample
1
2
3
If the display shows a whole number, then the original
number is a square number. For example, 576 is a square
number because using the square-root key displays a
whole number—24.
4
5
Name
If the display shows a decimal, then the original number is not
a square number. For example, 794 is not a square number
because using the square-root key displays a decimal—
28.178006 (rounded to 6 decimal places).
Date
Time
1 2
4 3
Top-It Record Sheet
Round
>, <, =
Player 1
Player 2
Sample
1
2
Ask students to check the remaining numbers in Problem 2.
Partners complete the remaining problems on journal page 23.
3
4
5
2 Ongoing Learning & Practice
▶ Playing Multiplication Top-It
Math Masters, p. 493
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PARTNER
ACTIVITY
(Extended-Facts Version)
(Student Reference Book, p. 334; Math Masters, p. 493)
Students apply their knowledge of basic multiplication facts to
extended facts by playing Multiplication Top-It (Extended-Facts
Version). Students use the same rules as described on Student
Reference Book, page 334; however, they attach a zero to the first
card drawn and multiply by the second card drawn. For example,
suppose 5 is the first number drawn; 7 is the second number
drawn. The student would compute: 5 ∗ 10 = 50; 50 ∗ 7 = 350.
Student Page
Date
Time
LESSON
Math Boxes
18
Math Masters
Ongoing Assessment:
Page 493
Recognizing Student Achievement
Use the Top-It Record Sheet (Math Masters, page 493) to assess students’
ability to solve and compare multiplication extended fact problems. Have the
class record and compare 70 ∗ 8 and 50 ∗ 9 for the sample record. Partners
record their first five rounds. Students are making adequate progress if they
correctly solve and compare all five extended facts. Some students may be able
to solve and compare problems with both factors multiplied by 10: 70 ∗ 80.
1.
Write < or >.
a.
3.8
b.
0.4
c.
6.24
d.
0.05
e.
7.12
>
>
>
<
<
2.
49,573
c.
2,601,458
d.
300,297
e.
599,999
0.5
7.2
9,000
50,000
2,601,000
300,000
600,000
9 32 33
3.
5.
4 249
List all the factors of 64.
4.
1, 2, 4, 8, 16, 32, 64
In the morning, I need 30 minutes to
shower and dress, 15 minutes to eat, and
another 15 minutes to ride my bike to
school. School begins at 8:30 A.M. What is
the latest time I can get up and still get to
school on time?
7:30 A.M.
244 245
Subtract. Show your work.
a.
INDEPENDENT
ACTIVITY
8,692
b.
6.08
10 12
a.
0.30
[Operations and Computation Goal 2]
▶ Math Boxes 1 8
Round each number to the
nearest thousand.
0.83
777
− 259
518
b.
555
− 125
430
c.
5,009
− 188
d.
4,821
8,435
− 997
15–17
7,438
(Math Journal 1, p. 24)
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 1-6. The skill in Problem 5
previews Unit 2 content.
Math Journal 1, p. 24
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Lesson 1 8
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Study Link Master
Name
Date
STUDY LINK
Time
Factor Rainbows, Squares, and Square Roots
18
䉬
List all the factors of each square number. Make a factor rainbow to check
your work. Then fill in the missing numbers.
1.
Reminder: In a factor rainbow, the
product of each connected factor pair
should be equal to the number itself.
For example, the factor rainbow for
16 looks like this:
1
1 º 16 16
2
4
271
8 16
2 º 8 16
4 º 4 16
Example:
12 4
1, 2, 4
4:
2
2
4 The square root of 4 is
9:
3 29
36:
The square root of 9 is
1 2 3 4 6 9 12 18 36
6 2 36 The square root of 36 is 6 .
5.
The square root of 25 is
Do all square numbers have an odd number of factors?
2.
Yes
Unsquare each number. The result is its square root. Do not use the
square root key
on your calculator.
11
3.
2
121
4.
The square root of 121 is
3.
1, 2, 3, 4, 6, 9, 12, 18, 36
▶ Study Link 1 8
INDEPENDENT
ACTIVITY
(Math Masters, p. 22)
1 5 25
5 2 25
1, 3, 9
1 3 9
2.
1, 5, 25
25:
Writing/Reasoning Have students write a response to the
following: Was Jason correct when he said that 64 is a
prime number in Problem 3? Explain your answer.
Sample answer: Jason was not correct. The factors of 64 are 1, 2, 4,
8, 16, 32, and 64. Because it has more than two factors, it is a
composite number. A prime number has only two factors.
11 .
50
2
2,500
The square root of 2,500 is
Home Connection Students list all the factors of the first
4 square numbers, write numbers in exponential notation,
and identify square roots.
50 .
Practice
4,318
1,901
5.
8.
36
85
6.
6,219
50 6 ∑ 8 R2
7.
3,060
9.
333 291 2,852
5
14,260
42
3 Differentiation Options
Math Masters, p. 22
ENRICHMENT
▶ Comparing Numbers with
PARTNER
ACTIVITY
15–30 Min
Their Squares
(Math Masters, p. 23)
To further explore factoring numbers, have students investigate
the relationship between numbers and their squares. Ask students
to think about the following question as they work the problems
for this activity: When you square a number, will the result be
greater than, less than, or equal to the number?
Teaching Master
Name
Date
LESSON
Comparing Numbers with Their Squares
18
䉬
1. a.
Unsquare the number 1.
b.
Unsquare the number 0.
2. a.
2
1
2
0
Is 5 greater than or less than 1?
c.
Is 52 greater than or less than 5?
c.
4. a.
Greater than
25
52 b.
5.
1
0
b.
3. a.
Time
Greater than
Less than
Is 0.50 greater than or less than 1?
Use your calculator. 0.50 0.25
Less than
Is 0.50 greater than or less than 0.50?
Guide students to recognize that squaring a number does not
necessarily result in a number that is greater than the original
number. For example, both 0 and 1 are equal to their squares.
(See Problem 1.) Ask students what they noticed about the
numbers and relationships they found in Problem 3. The number
0.50 is a decimal; the square was smaller. Explain that the square
of a number that is greater than 0, but less than 1, is always less
than the original number. Ask volunteers to suggest other
numbers between 0 and 1 for partners to square.
2
2
When you square a number, is the result
always greater than the number you started with?
EXTRA PRACTICE
No
Yes
Yes
b.
Can it be less?
c.
Can it be the same?
Write 3 true statements about squaring and unsquaring numbers.
Answers vary.
▶ 5-Minute Math
SMALL-GROUP
ACTIVITY
5–15 Min
To offer students more experience with using the square root sign,
see 5-Minute Math, page 108.
Math Masters, p. 23
56
Unit 1
Number Theory
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Name
Date
Time
1 2
4 3
Top-It Record Sheet
Round
Player 1
>, <, =
Player 2
Sample
1
2
3
4
5
Name
Date
Time
1 2
4 3
Top-It Record Sheet
Copyright © Wright Group/McGraw-Hill
Round
Player 1
>, <, =
Player 2
Sample
1
2
3
4
5
493
EM3cuG5MM_U03_067-101.indd 493
11/10/10 4:36 PM